#set document(title: "4.4 Uniform and Nonuniform Circular Motion", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 4.4#h(0.6em)Uniform and Nonuniform Circular Motion Uniform circular motion is a specific type of motion in which an object travels in a circle with a constant speed. For example, any point on a propeller spinning at a constant rate is executing uniform circular motion. Other examples are the second, minute, and hour hands of a watch. It is remarkable that points on these rotating objects are actually accelerating, although the rotation rate is a constant. To see this, we must analyze the motion in terms of vectors. === Centripetal Acceleration In one-dimensional kinematics, objects with a constant speed have zero acceleration. However, in two- and three-dimensional kinematics, even if the speed is a constant, a particle can have acceleration if it moves along a curved trajectory such as a circle. In this case the velocity vector is changing, or #math.equation(block: false, alt: "d v → / d t not equal to 0 .")[$d arrow(v) "/" d t ≠ 0 .$] This is shown. As the particle moves counterclockwise in time #math.equation(block: false, alt: "Δ t")[$"Δ" t$] on the circular path, its position vector moves from #math.equation(block: false, alt: "r → open parenthesis t close parenthesis")[$arrow(r) ( t )$] to #math.equation(block: false, alt: "r → open parenthesis t plus Δ t close parenthesis .")[$arrow(r) ( t + "Δ" t ) .$] The velocity vector has constant magnitude and is tangent to the path as it changes from #math.equation(block: false, alt: "v → open parenthesis t close parenthesis")[$arrow(v) ( t )$] to #math.equation(block: false, alt: "v → open parenthesis t plus Δ t close parenthesis ,")[$arrow(v) ( t + "Δ" t ) ,$] changing its direction only. Since the velocity vector #math.equation(block: false, alt: "v → open parenthesis t close parenthesis")[$arrow(v) ( t )$] is perpendicular to the position vector #math.equation(block: false, alt: "r → open parenthesis t close parenthesis ,")[$arrow(r) ( t ) ,$] the triangles formed by the position vectors and #math.equation(block: false, alt: "Δ r → ,")[$"Δ" arrow(r) ,$] and the velocity vectors and #math.equation(block: false, alt: "Δ v →")[$"Δ" arrow(v)$] are similar. Furthermore, since #math.equation(block: false, alt: "| r → open parenthesis t close parenthesis | equals | r → open parenthesis t plus Δ t close parenthesis |")[$| arrow(r) ( t ) | = | arrow(r) ( t + "Δ" t ) |$] and #math.equation(block: false, alt: "| v → open parenthesis t close parenthesis | equals | v → open parenthesis t plus Δ t close parenthesis | ,")[$| arrow(v) ( t ) | = | arrow(v) ( t + "Δ" t ) | ,$] the two triangles are isosceles. From these facts we can make the assertion #math.equation(block: false, alt: "the fraction Δ v over v equals the fraction Δ r over r")[$frac("Δ" v, v) = frac("Δ" r, r)$] or #math.equation(block: false, alt: "Δ v equals the fraction v over r Δ r .")[$"Δ" v = frac(v, r) "Δ" r .$] #figure(figph[Figure a shows a circle with center at point C. We are shown radius r of t and radius r of t, which are an angle Delta theta apart, and the chord length delta r connecting the ends of the two radii. Vectors r of t, r of t plus delta t, and delta r form a triangle. At the tip of vector r of t, the velocity is shown as v of t and points up and to the right, tangent to the circle. . At the tip of vector r of t plus delta t, the velocity is shown as v of t plus delta t and points up and to the left, tangent to the circle. Figure b shows the vectors v of t and v of t plus delta t with their tails together, and the vector delta v from the tip of v of t to the tip of v of t plus delta t. These three vectors form a triangle. The angle between the v of t and v of t plus delta t is theta.], alt: "Figure a shows a circle with center at point C. We are shown radius r of t and radius r of t, which are an angle Delta theta apart, and the chord length delta r connecting the ends of the two radii. Vectors r of t, r of t plus delta t, and delta r form a triangle. At the tip of vector r of t, the velocity is shown as v of t and points up and to the right, tangent to the circle. . At the tip of vector r of t plus delta t, the velocity is shown as v of t plus delta t and points up and to the left, tangent to the circle. Figure b shows the vectors v of t and v of t plus delta t with their tails together, and the vector delta v from the tip of v of t to the tip of v of t plus delta t. These three vectors form a triangle. The angle between the v of t and v of t plus delta t is theta.", caption: [(a) A particle is moving in a circle at a constant speed, with position and velocity vectors at times #math.equation(block: false, alt: "t")[$t$] and #math.equation(block: false, alt: "t plus Δ t .")[$t + "Δ" t .$] (b) Velocity vectors forming a triangle. The two triangles in the figure are similar. The vector #math.equation(block: false, alt: "Δ v →")[$"Δ" arrow(v)$] points toward the center of the circle in the limit #math.equation(block: false, alt: "Δ t → 0 .")[$"Δ" t → 0 .$]]) We can find the magnitude of the acceleration from #math.equation(block: true, alt: "a equals lim Δ t → 0 open parenthesis the fraction Δ v over Δ t close parenthesis equals the fraction v over r open parenthesis lim Δ t → 0 the fraction Δ r over Δ t close parenthesis equals the fraction v squared over r .")[$a = limits("lim")_("Δ" t → 0) ( frac("Δ" v, "Δ" t) ) = frac(v, r) ( limits("lim")_("Δ" t → 0) frac("Δ" r, "Δ" t) ) = frac(v^(2), r) .$] The direction of the acceleration can also be found by noting that as #math.equation(block: false, alt: "Δ t")[$"Δ" t$] and therefore #math.equation(block: false, alt: "Δ θ")[$"Δ" θ$] approach zero, the vector #math.equation(block: false, alt: "Δ v →")[$"Δ" arrow(v)$] approaches a direction perpendicular to #math.equation(block: false, alt: "v → .")[$arrow(v) .$] In the limit #math.equation(block: false, alt: "Δ t → 0 ,")[$"Δ" t → 0 ,$]#math.equation(block: false, alt: "Δ v →")[$"Δ" arrow(v)$] is perpendicular to #math.equation(block: false, alt: "v → .")[$arrow(v) .$] Since #math.equation(block: false, alt: "v →")[$arrow(v)$] is tangent to the circle, the acceleration #math.equation(block: false, alt: "d v → / d t")[$d arrow(v) "/" d t$] points toward the center of the circle. Summarizing, a particle moving in a circle at a constant speed has an acceleration with magnitude #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "a sub c equals the fraction v squared over r .")[$a_("c") = frac(v^(2), r) .$] ] The direction of the acceleration vector is toward the center of the circle . This is a radial acceleration and is called the #strong[centripetal acceleration], which is why we give it the subscript c. The word #emph[centripetal] comes from the Latin words #emph[centrum] (meaning “center”) and #emph[petere] (meaning “to seek”), and thus takes the meaning “center seeking.” #figure(figph[A circle is shown with a purple arrow labeled as vector a sub C pointing radially inward and a green arrow tangent to the circle and labeled v. The arrows are shown with their tails at the same point on the circle.], alt: "A circle is shown with a purple arrow labeled as vector a sub C pointing radially inward and a green arrow tangent to the circle and labeled v. The arrows are shown with their tails at the same point on the circle.", caption: [The centripetal acceleration vector points toward the center of the circular path of motion and is an acceleration in the radial direction. The velocity vector is also shown and is tangent to the circle.]) Let’s investigate some examples that illustrate the relative magnitudes of the velocity, radius, and centripetal acceleration. #examplebox("Example 1")[Creating an Acceleration of 1 #emph[g]][ A jet is flying at 134.1 m/s along a straight line and makes a turn along a circular path level with the ground. What does the radius of the circle have to be to produce a centripetal acceleration of 1 #emph[g] on the pilot and jet toward the center of the circular trajectory? Strategy Given the speed of the jet, we can solve for the radius of the circle in the expression for the centripetal acceleration. Solution Set the centripetal acceleration equal to the acceleration of gravity: #math.equation(block: false, alt: "9.8 m/s squared equals v squared / r .")[$9.8 attach(#h(0.2em) "m/s", t: 2) = v^(2) "/" r .$] Solving for the radius, we find #math.equation(block: true, alt: "r equals the fraction open parenthesis 134.1 m / s close parenthesis squared over 9.8 m / s squared equals 1835 m equals 1.835 km .")[$r = frac(( 134.1 #h(0.2em) "m" "/" attach("s" ), t: 2), 9.8 #h(0.2em) "m" "/" "s"^(2)) = 1835 #h(0.2em) "m" = 1.835 #h(0.2em) "km" "."$] Significance To create a greater acceleration than #emph[g] on the pilot, the jet would either have to decrease the radius of its circular trajectory or increase its speed on its existing trajectory or both. ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ A flywheel has a radius of 20.0 cm. What is the speed of a point on the edge of the flywheel if it experiences a centripetal acceleration of #math.equation(block: false, alt: "900.0 cm / s squared ?")[$900.0 #h(0.2em) attach("cm" "/" "s", t: 2) ?$] #solutionbox[ 134.0 cm/s ] ] Centripetal acceleration can have a wide range of values, depending on the speed and radius of curvature of the circular path. Typical centripetal accelerations are given in the following table. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Object], [Centripetal Acceleration (m/s#super[2] or factors of #emph[g])]), [Earth around the Sun], [#math.equation(block: false, alt: "5.93 times 10 to the power −3")[$5.93 #h(0.2em) × #h(0.2em) 10^(−3)$]], [Moon around the Earth], [#math.equation(block: false, alt: "2.73 times 10 to the power −3")[$2.73 #h(0.2em) × #h(0.2em) 10^(−3)$]], [Satellite in geosynchronous orbit], [0.233], [Outer edge of a CD when playing], [#math.equation(block: false, alt: "5.78")[$5.78$]], [Jet in a barrel roll], [(2–3 #emph[g])], [Roller coaster], [(5 #emph[g])], [Electron orbiting a proton in a simple Bohr model of the atom], [#math.equation(block: false, alt: "9.0 times 10 to the power 22")[$9.0 #h(0.2em) × #h(0.2em) 10^(22)$]], )) === Equations of Motion for Uniform Circular Motion A particle executing circular motion can be described by its position vector #math.equation(block: false, alt: "r → open parenthesis t close parenthesis .")[$arrow(r) ( t ) .$] shows a particle executing circular motion in a counterclockwise direction. As the particle moves on the circle, its position vector sweeps out the angle #math.equation(block: false, alt: "θ")[$θ$] with the #emph[x-]axis. Vector #math.equation(block: false, alt: "r → open parenthesis t close parenthesis")[$arrow(r) ( t )$] making an angle #math.equation(block: false, alt: "θ")[$θ$] with the #emph[x-]axis is shown with its components along the #emph[x]- and #emph[y]-axes. The magnitude of the position vector is #math.equation(block: false, alt: "A equals | r → open parenthesis t close parenthesis |")[$A = | arrow(r) ( t ) |$] and is also the radius of the circle, so that in terms of its components, #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "r → open parenthesis t close parenthesis equals A cos ω t i ^ plus A sin ω t j ^ .")[$arrow(r) ( t ) = A #h(0.1em) "cos" #h(0.1em) ω t hat(i) + A #h(0.1em) "sin" #h(0.2em) ω t hat(j) .$] ] Here, #math.equation(block: false, alt: "ω")[$ω$] is a constant called the #strong[angular frequency] of the particle. The angular frequency has units of radians (rad) per second and is simply the number of radians of angular measure through which the particle passes per second. The angle #math.equation(block: false, alt: "θ")[$θ$] that the position vector has at any particular time is #math.equation(block: false, alt: "ω t")[$ω t$]. If #emph[T] is the period of motion, or the time to complete one revolution (#math.equation(block: false, alt: "2 π")[$2 π$] rad), then #math.equation(block: true, alt: "ω equals the fraction 2 π over T .")[$ω = frac(2 π, T) .$] #figure(figph[A circle radius r, centered on the origin of an x y coordinate system is shown. Radius r of t is a vector from the origin to a point on the circle and is at an angle of theta equal to omega t to the horizontal. The x component of vector r is the magnitude of r of t times cosine of omega t. The y component of vector r is the magnitude of r of t times sine of omega t. The circulation is counterclockwise around the circle.], alt: "A circle radius r, centered on the origin of an x y coordinate system is shown. Radius r of t is a vector from the origin to a point on the circle and is at an angle of theta equal to omega t to the horizontal. The x component of vector r is the magnitude of r of t times cosine of omega t. The y component of vector r is the magnitude of r of t times sine of omega t. The circulation is counterclockwise around the circle.", caption: [The position vector for a particle in circular motion with its components along the #emph[x]- and #emph[y]-axes. The particle moves counterclockwise. Angle #math.equation(block: false, alt: "θ")[$θ$] is the angular frequency #math.equation(block: false, alt: "ω")[$ω$] in radians per second multiplied by #emph[t].]) Velocity and acceleration can be obtained from the position function by differentiation: #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "v → open parenthesis t close parenthesis equals the fraction d r → open parenthesis t close parenthesis over d t equals − A ω sin ω t i ^ plus A ω cos ω t j ^ .")[$arrow(v) ( t ) = frac(d arrow(r) ( t ), d t) = "−" A ω #h(0.1em) "sin" #h(0.2em) ω t hat(i) + A ω #h(0.1em) "cos" #h(0.2em) ω t hat(j) .$] ] It can be shown from that the velocity vector is tangential to the circle at the location of the particle, with magnitude #math.equation(block: false, alt: "A ω .")[$A ω .$] Similarly, the acceleration vector is found by differentiating the velocity: #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "a → open parenthesis t close parenthesis equals the fraction d v → open parenthesis t close parenthesis over d t equals − A ω squared cos ω t i ^ minus A ω squared sin ω t j ^ .")[$arrow(a) ( t ) = frac(d arrow(v) ( t ), d t) = "−" A ω^(2) #h(0.2em) "cos" #h(0.2em) ω t hat(i) − A ω^(2) #h(0.2em) "sin" #h(0.2em) ω t hat(j) .$] ] From this equation we see that the acceleration vector has magnitude #math.equation(block: false, alt: "A ω squared")[$A ω^(2)$] and is directed opposite the position vector, toward the origin, because #math.equation(block: false, alt: "a → open parenthesis t close parenthesis equals − ω squared r → open parenthesis t close parenthesis .")[$arrow(a) ( t ) = "−" ω^(2) arrow(r) ( t ) .$] #examplebox("Example 2")[Circular Motion of a Proton][ A proton has speed #math.equation(block: false, alt: "5 times 10 to the power 6 m/s")[$5 #h(0.2em) × #h(0.2em) 10^(6) #h(0.2em) "m/s"$] and is moving in a circle in the #emph[xy] plane of radius #emph[r] = 0.175 m. What is its position in the #emph[xy] plane at time #math.equation(block: false, alt: "t equals 2.0 times 10 to the power −7 s equals 200 ns?")[$t = 2.0 #h(0.2em) × #h(0.2em) 10^(−7) #h(0.2em) "s" = 200 #h(0.2em) "ns?"$] At #emph[t] = 0, the position of the proton is #math.equation(block: false, alt: "0.175 m i ^")[$0.175 #h(0.2em) "m" hat(i)$] and it circles counterclockwise. Sketch the trajectory. Solution According to , #math.equation(block: true, alt: "Average speed equals the fraction Total distance over Elapsed time .")[$"Average speed" = frac("Total distance", "Elapsed time") .$] Since the period #emph[T] is the time it takes an object to go once around a circle, and the distance around a circle is 2#emph[πr], we have: #math.equation(block: true, alt: "v equals the fraction 2 π r over T .")[$v = frac(2 π r, T) .$] From the given data, the proton has period and angular frequency: #math.equation(block: true, alt: "T equals the fraction 2 π r over v equals the fraction 2 π open parenthesis 0.175 m close parenthesis over 5.0 times 10 to the power 6 m / s equals 2.20 times 10 to the power −7 s")[$T = frac(2 π r, v) = frac(2 π ( 0.175 #h(0.2em) "m" ), 5.0 #h(0.2em) × #h(0.2em) 10^(6) #h(0.2em) "m" "/" "s") = 2.20 #h(0.2em) × #h(0.2em) 10^(−7) #h(0.2em) "s"$] #math.equation(block: true, alt: "ω equals the fraction 2 π over T equals the fraction 2 π over 2.20 times 10 to the power −7 s equals 2.856 times 10 to the power 7 rad / s .")[$ω = frac(2 π, T) = frac(2 π, 2.20 #h(0.2em) × #h(0.2em) 10^(−7) #h(0.2em) "s") = 2.856 #h(0.2em) × #h(0.2em) 10^(7) #h(0.2em) "rad" "/" "s" .$] The position of the particle at #math.equation(block: false, alt: "t equals 2.0 times 10 to the power −7 s")[$t = 2.0 #h(0.2em) × #h(0.2em) 10^(−7) #h(0.2em) "s"$] with #emph[A] = 0.175 m is #math.equation(block: true, alt: "r → open parenthesis 2.0 times 10 to the power −7 s close parenthesis, equals A cos ω open parenthesis 2.0 times 10 to the power −7 s close parenthesis i ^ plus A sin ω open parenthesis 2.0 times 10 to the power −7 s close parenthesis j ^ m; equals 0.175 cos [ open parenthesis 2.856 times 10 to the power 7 rad / s close parenthesis open parenthesis 2.0 times 10 to the power −7 s close parenthesis ] i ^; plus 0.175 sin [ open parenthesis 2.856 times 10 to the power 7 rad / s close parenthesis open parenthesis 2.0 times 10 to the power −7 s close parenthesis ] j ^ m; equals 0.175 cos open parenthesis 5.712 rad close parenthesis i ^ plus 0.175 sin open parenthesis 5.712 rad close parenthesis j ^ equals 0.147 i ^ minus 0.095 j ^ m .")[$arrow(r) ( 2.0 #h(0.2em) × #h(0.2em) 10^(−7) "s" ) & = A #h(0.2em) "cos" #h(0.2em) ω ( 2.0 #h(0.2em) × #h(0.2em) 10^(−7) #h(0.2em) "s" ) hat(i) + A #h(0.2em) "sin" #h(0.2em) ω ( 2.0 #h(0.2em) × #h(0.2em) 10^(−7) #h(0.2em) "s" ) hat(j) #h(0.2em) "m" \ & = 0.175 "cos" [ ( 2.856 #h(0.2em) × #h(0.2em) 10^(7) #h(0.2em) "rad" "/" "s" ) ( 2.0 #h(0.2em) × #h(0.2em) 10^(−7) #h(0.2em) "s" ) ] hat(i) \ & + 0.175 "sin" [ ( 2.856 #h(0.2em) × #h(0.2em) 10^(7) #h(0.2em) "rad" "/" "s" ) ( 2.0 #h(0.2em) × #h(0.2em) 10^(−7) #h(0.2em) "s" ) ] hat(j) #h(0.2em) "m" \ & = 0.175 "cos" ( 5.712 #h(0.2em) "rad" ) hat(i) + 0.175 "sin" ( 5.712 #h(0.2em) "rad" ) hat(j) = 0.147 hat(i) − 0.095 hat(j) #h(0.2em) "m" .$] From this result we see that the proton is located slightly below the #emph[x]-axis. This is shown. #figure(figph[A graph of y position as a function of x position is shown. Both x and y are measured in meters and run from -0.2 to 0.2. A proton is moving in a counterclockwise circle centered on the origin is shown at 11 different times. At t = 0 s the particle is at x = 0.175 m and y = 0. At t = 200 nanoseconds, the particle is at a position given by vector 0.147 I hat minus 0.095 j hat meters.], alt: "A graph of y position as a function of x position is shown. Both x and y are measured in meters and run from -0.2 to 0.2. A proton is moving in a counterclockwise circle centered on the origin is shown at 11 different times. At t = 0 s the particle is at x = 0.175 m and y = 0. At t = 200 nanoseconds, the particle is at a position given by vector 0.147 I hat minus 0.095 j hat meters.", caption: [Position vector of the proton at #math.equation(block: false, alt: "t equals 2.0 times 10 to the power −7 s equals 200 ns .")[$t = 2.0 #h(0.2em) × #h(0.2em) 10^(−7) "s" = 200 #h(0.2em) "ns" "."$] The trajectory of the proton is shown. The angle through which the proton travels along the circle is 5.712 rad, which a little less than one complete revolution.]) Significance We picked the initial position of the particle to be on the #emph[x-]axis. This was completely arbitrary. If a different starting position were given, we would have a different final position at #emph[t] = 200 ns. ] === Nonuniform Circular Motion Circular motion does not have to be at a constant speed. A particle can travel in a circle and speed up or slow down, showing an acceleration in the direction of the motion. In uniform circular motion, the particle executing circular motion has a constant speed and the circle is at a fixed radius. If the speed of the particle is changing as well, then we introduce an additional acceleration in the direction tangential to the circle. Such accelerations occur at a point on a top that is changing its spin rate, or any accelerating rotor. In Displacement and Velocity Vectors we showed that centripetal acceleration is the time rate of change of the direction of the velocity vector. If the speed of the particle is changing, then it has a #strong[tangential acceleration] that is the time rate of change of the magnitude of the velocity: #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "a sub T equals the fraction d | v → | over d t .")[$a_("T") = frac(d | arrow(v) |, d t) .$] ] The direction of tangential acceleration is tangent to the circle whereas the direction of centripetal acceleration is radially inward toward the center of the circle. Thus, a particle in circular motion with a tangential acceleration has a #strong[total acceleration] that is the vector sum of the centripetal and tangential accelerations: #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: true, alt: "a → equals a → sub c plus a → sub T .")[$arrow(a) = arrow(a)_("c") + arrow(a)_("T") .$] ] The acceleration vectors are shown. Note that the two acceleration vectors #math.equation(block: false, alt: "a → sub c")[$arrow(a)_("c")$] and #math.equation(block: false, alt: "a → sub T")[$arrow(a)_("T")$] are perpendicular to each other, with #math.equation(block: false, alt: "a → sub c")[$arrow(a)_("c")$] in the radial direction and #math.equation(block: false, alt: "a → sub T")[$arrow(a)_("T")$] in the tangential direction. The total acceleration #math.equation(block: false, alt: "a →")[$arrow(a)$] points at an angle between #math.equation(block: false, alt: "a → sub c")[$arrow(a)_("c")$] and #math.equation(block: false, alt: "a → sub T .")[$arrow(a)_("T") .$] #figure(figph[The acceleration of a particle on a circle is shown along with its radial and tangential components. The centripetal acceleration a sub c points radially toward the center of the circle. The tangential acceleration a sub T is tangential to the circle at the particle’s position. The total acceleration is the vector sum of the tangential and centripetal accelerations, which are perpendicular.], alt: "The acceleration of a particle on a circle is shown along with its radial and tangential components. The centripetal acceleration a sub c points radially toward the center of the circle. The tangential acceleration a sub T is tangential to the circle at the particle’s position. The total acceleration is the vector sum of the tangential and centripetal accelerations, which are perpendicular.", caption: [The centripetal acceleration points toward the center of the circle. The tangential acceleration is tangential to the circle at the particle’s position. The total acceleration is the vector sum of the tangential and centripetal accelerations, which are perpendicular.]) #examplebox("Example 3")[Total Acceleration during Circular Motion][ A particle moves in a circle of radius #emph[r] = 2.0 m. During the time interval from #emph[t] = 1.5 s to #emph[t] = 4.0 s its speed varies with time according to #math.equation(block: true, alt: "v open parenthesis t close parenthesis equals c sub 1 minus the fraction c sub 2 over t squared , c sub 1 equals 4.0 m / s, c sub 2 equals 6.0 m times s .")[$v ( t ) = c_(1) − frac(c_(2), t^(2)) , #h(0.5em) c_(1) = 4.0 #h(0.2em) "m" "/" "s," #h(0.5em) c_(2) = 6.0 #h(0.2em) "m" · "s" "."$] What is the total acceleration of the particle at #emph[t] = 2.0 s? Strategy We are given the speed of the particle and the radius of the circle, so we can calculate centripetal acceleration easily. The direction of the centripetal acceleration is toward the center of the circle. We find the magnitude of the tangential acceleration by taking the derivative with respect to time of #math.equation(block: false, alt: "| v open parenthesis t close parenthesis |")[$| v ( t ) |$] using and evaluating it at #emph[t] = 2.0 s. We use this and the magnitude of the centripetal acceleration to find the total acceleration. Solution Centripetal acceleration is #math.equation(block: true, alt: "v open parenthesis 2.0 s close parenthesis equals open parenthesis 4.0 minus the fraction 6.0 over open parenthesis 2.0 close parenthesis squared close parenthesis m / s equals 2.5 m / s")[$v ( 2.0 "s" ) = ( 4.0 − frac(6.0, attach(( 2.0 ), t: 2)) ) "m" "/" "s" = 2.5 #h(0.2em) "m" "/" "s"$] #math.equation(block: true, alt: "a sub c equals the fraction v squared over r equals the fraction open parenthesis 2.5 m / s close parenthesis squared over 2.0 m equals 3.1 m / s squared")[$a_("c") = frac(v^(2), r) = frac(( 2.5 #h(0.2em) "m" "/" attach("s" ), t: 2), 2.0 #h(0.2em) "m") = 3.1 #h(0.2em) "m" "/" "s"^(2)$] directed toward the center of the circle. Tangential acceleration is #math.equation(block: true, alt: "a sub T equals | the fraction d v → over d t | equals the fraction 2 c sub 2 over t cubed equals the fraction 12.0 over open parenthesis 2.0 close parenthesis cubed m / s squared equals 1.5 m / s squared .")[$a_("T") = | frac(d arrow(v), d t) | = frac(2 c_(2), t^(3)) = frac(12.0, attach(( 2.0 ), t: 3)) "m" "/" "s"^(2) = 1.5 #h(0.2em) "m" "/" "s"^(2) .$] Total acceleration is #math.equation(block: true, alt: "| a → | equals the square root of 3.1 squared plus 1.5 squared m / s squared equals 3.44 m / s squared")[$| arrow(a) | = sqrt(3.1^(2) + 1.5^(2)) "m" "/" "s"^(2) = 3.44 #h(0.2em) "m" "/" "s"^(2)$] and #math.equation(block: false, alt: "θ equals tan to the power −1 the fraction 3.1 over 1.5 equals 64 °")[$θ = "tan"^(−1) frac(3.1, 1.5) = 64 "°"$] from the tangent to the circle. #figure(figph[The acceleration of a particle on a circle is shown along with its radial and tangential components. The centripetal acceleration a sub c points radially toward the center of the circle and has magnitude 3.1 meters per second squared. The tangential acceleration a sub T is tangential to the circle at the particle’s position and has magnitude 1.5 meters per second squared. The angle between the total acceleration a and the tangential acceleration a sub T is 64 degrees.], alt: "The acceleration of a particle on a circle is shown along with its radial and tangential components. The centripetal acceleration a sub c points radially toward the center of the circle and has magnitude 3.1 meters per second squared. The tangential acceleration a sub T is tangential to the circle at the particle’s position and has magnitude 1.5 meters per second squared. The angle between the total acceleration a and the tangential acceleration a sub T is 64 degrees.", caption: [The tangential and centripetal acceleration vectors. The net acceleration #math.equation(block: false, alt: "a →")[$arrow(a)$] is the vector sum of the two accelerations.]) Significance The directions of centripetal and tangential accelerations can be described more conveniently in terms of a polar coordinate system, with unit vectors in the radial and tangential directions. This coordinate system, which is used for motion along curved paths, is discussed in detail later in the book. ] === Summary - Uniform circular motion is motion in a circle at constant speed. - Centripetal acceleration #math.equation(block: false, alt: "a → sub C")[$arrow(a)_("C")$] is the acceleration a particle must have to follow a circular path. Centripetal acceleration always points toward the center of rotation and has magnitude #math.equation(block: false, alt: "a sub C equals v squared / r .")[$a_("C") = v^(2) "/" r .$] - Nonuniform circular motion occurs when there is tangential acceleration of an object executing circular motion such that the speed of the object is changing. This acceleration is called tangential acceleration #math.equation(block: false, alt: "a → sub T .")[$arrow(a)_("T") .$] The magnitude of tangential acceleration is the time rate of change of the magnitude of the velocity. The tangential acceleration vector is tangential to the circle, whereas the centripetal acceleration vector points radially inward toward the center of the circle. The total acceleration is the vector sum of tangential and centripetal accelerations. - An object executing uniform circular motion can be described with equations of motion. The position vector of the object is #math.equation(block: false, alt: "r → open parenthesis t close parenthesis equals A cos ω t i ^ plus A sin ω t j ^ ,")[$arrow(r) ( t ) = A #h(0.2em) "cos" #h(0.2em) ω t hat(i) + A #h(0.2em) "sin" #h(0.2em) ω t hat(j) ,$] where #emph[A] is the magnitude #math.equation(block: false, alt: "| r → open parenthesis t close parenthesis | ,")[$| arrow(r) ( t ) | ,$] which is also the radius of the circle, and #math.equation(block: false, alt: "ω")[$ω$] is the angular frequency. === Conceptual Questions Can centripetal acceleration change the speed of a particle undergoing circular motion? Can tangential acceleration change the speed of a particle undergoing circular motion? #solutionbox[ yes ] === Problems A flywheel is rotating at 30 rev/s. What is the total angle, in radians, through which a point on the flywheel rotates in 40 s? A particle travels in a circle of radius 10 m at a constant speed of 20 m/s. What is the magnitude of the acceleration? #solutionbox[ #math.equation(block: true, alt: "a sub C equals 40 m / s squared")[$a_("C") = 40 #h(0.2em) "m" "/" "s"^(2)$] ] Cam Newton of the Carolina Panthers throws a perfect football spiral at 8.0 rev/s. The radius of a pro football is 8.5 cm at the middle of the short side. What is the centripetal acceleration of the laces on the football? A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00-m radius, at how many revolutions per minute are the riders subjected to a centripetal acceleration equal to that of gravity? #solutionbox[ #math.equation(block: false, alt: "a sub C equals the fraction v squared over r ⇒ v squared equals r a sub C equals 78.4 , v equals 8.85 m / s")[$a_("C") = frac(v^(2), r) ⇒ v^(2) = r #h(0.5em) a_("C") = 78.4 , #h(0.5em) v = 8.85 #h(0.2em) "m" "/" "s"$] #linebreak() #math.equation(block: false, alt: "T equals 5.68 s,")[$T = 5.68 #h(0.2em) "s,"$] which is #math.equation(block: false, alt: "0.176 rev / s equals 10.6 rev / min")[$0.176 #h(0.2em) "rev" "/" "s" = 10.6 #h(0.2em) "rev" "/" "min"$] ] A runner taking part in the 200-m dash must run around the end of a track that has a circular arc with a radius of curvature of 30.0 m. The runner starts the race at a constant speed. If she completes the 200-m dash in 23.2 s and runs at constant speed throughout the race, what is her centripetal acceleration as she runs the curved portion of the track? What is the acceleration of Venus toward the Sun, assuming a circular orbit? #solutionbox[ Venus is 108.2 million km from the Sun and has an orbital period of 0.6152 y. #linebreak() #math.equation(block: false, alt: "r equals 1.082 times 10 to the power 11 m T equals 1.94 times 10 to the power 7 s")[$r = 1.082 #h(0.2em) × #h(0.2em) 10^(11) #h(0.2em) "m" #h(0.5em) T = 1.94 #h(0.2em) × #h(0.2em) "10"^(7) #h(0.2em) "s"$] #linebreak() #math.equation(block: false, alt: "v equals 3.5 times 10 to the power 4 m/s, a sub C equals 1.135 times 10 to the power −2 m/s squared")[$v = 3.5 #h(0.2em) × #h(0.2em) 10^(4) #h(0.2em) "m/s," #h(0.5em) "a"_("C") = 1.135 #h(0.2em) × #h(0.2em) 10^(−2) attach(#h(0.2em) "m/s", t: 2)$] ] An experimental jet rocket travels around Earth along its equator just above its surface. At what speed must the jet travel if the magnitude of its acceleration is #emph[g]? A fan is rotating at a constant 360.0 rev/min. What is the magnitude of the acceleration of a point on one of its blades 10.0 cm from the axis of rotation? #solutionbox[ #math.equation(block: true, alt: "360 rev / min equals 6 rev / s")[$360 #h(0.2em) "rev" "/" "min" = 6 #h(0.2em) "rev" "/" "s"$] #linebreak() #math.equation(block: true, alt: "v equals 3.8 m / s")[$v = 3.8 #h(0.2em) "m" "/" "s"$] #math.equation(block: true, alt: "a sub C equals 144. m / s squared")[$a_("C") = 144. #h(0.2em) "m" "/" "s"^(2)$] ] A point located on the second hand of a large clock has a radial acceleration of #math.equation(block: false, alt: "0.1 cm / s squared .")[$0.1 "cm" "/" "s"^(2) .$] How far is the point from the axis of rotation of the second hand?